Can you modify the surrounding curvature of a black hole while keeping the Schwarzschild radius constant?

Question

TL;DR: I'm attempting to make a black hole rendering, and I like the current size of the event-horizon but I want to reduce the curvature further away from it. But it seems like $g_{uv}$ only depends on $r_s$ given an object's current coordinates. Are there any other parameters I can change?

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Currently I have a spherical coordinate system $(ct, r, \theta, \phi)$ that extends 10 Unity meters in radius from the origin, with an $r_s$ around 0.5. When a ray intersects the spherical coordinate system, I express it's direction vector in terms of $e_r, e_\theta, e_\phi$, and calculate a component for $e_t$ by requiring $\frac{d}{d\lambda}\cdot\frac{d}{d\lambda} = 0$. Then from it's initial spherical coords/direction, I run 15,000 iterations of Euler's method on the geodesic equations, and convert back to Unity cartesian coordinates, dropping $\frac{d(ct)}{d\lambda}$ to send a new raycast. This is my current result:

I don't want to change the size of the event-horizon $r_s$, but I've am trying to reduce the curvature far away from the event horizon so that the region in pink smoothly connects to the surrounding flat space. As well, I am trying to get the white-light region outlined in orange closer to the event horizon, so it looks more like what I'm used to seeing.

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Both $g_{\theta\theta}$ and $g_{\phi\phi}$ are only dependent on the current coordinates.

$g_{tt} = 1-\frac{r_s}{r}$ and $g_{rr} = -g_{tt}^{-1}$, so it seems like the metric tensor is only a function of the Schwarzschild radius. So if I want to keep $r_s$ constant, there does not seem to be a parameter to reduce distance curvature.

I've tried putting random coefficients in the geodesic equations, but adjustments seem to have the effect of reducing distance curvature.

I've largely ignored $ct$, just including it for the sake of the geodesic equations and dropping it later. But I don't think changing it would have an affect, because $r_s$ is a function of $c^2$ already, which I am trying to hold constant. Nothing else in the code is in terms of G or M because $r_s$ is known in advanced.

Given that Newton's gravity is proportional to $\frac{1}{r^2}$, it would almost seem like you can scale the current geometry, but not change how it evolves over large distances.

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Are there any parameters that I'm not considering? Or is this not doable?

Thanks!

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Javier's answer worked:

Answer 1

The Schwarzschild metric only depends on $r_s$, there are no further parameters that you could tweak. I can think of two options:

• This is the most obvious one: you're seeing a sharp transition because you established a sharp transition between the curved and flat regions. You could just expand the curved region to cover the whole viewing area (which is more realistic anyway).

• Alternatively, you could modify the metric to artificially "flatten" it at some radius $R$. For example, you could take the function $$f(r) = 1 - \frac{r_s}{r}$$ that appears in the metric and replace it by $$\tilde{f}(r) = 1 - \frac{r_s}{r} e^{-r^2/R^2}.$$ For small $r$ this is very close to the Schwarzschild metric, and near $r \approx R$ it goes quickly to 1. You could replace the exponential by any function that goes from 1 to 0 at some given value.